As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers ( whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4 ), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table.

In other words, non-trivial propensities ( those that differ from 0 and 1 ) only exist for genuinely indeterministic experiments.

The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory.

For instance, if m is odd then n − m is also odd, and if m is even, then n − m is even, a non-trivial relation because ( besides 2 ) only odd numbers can be prime.

* A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotents.

But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.

A Table is in 4NF if and only if, for every one of its non-trivial multivalued dependencies X Y, X is a superkey — that is, X is either a candidate key or a superset thereof .< ref name =" Fagin ">" A relation schema R * is in fourth normal form ( 4NF ) if, whenever a nontrivial multivalued dependency X Y holds for R *, then so does the functional dependency X → A for every column name A of R *.

The third is the only example of a non-trivial connected sum with a geometric structure.

The concept of reference must not be confused with other values ( keys or identifiers ) that uniquely identify the data item, but give access to it only through a non-trivial lookup operation in some table data structure.

Kunen showed that 0 < sup >#</ sup > exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe L into itself.

Topologically, SO ( 3 ) is the real projective space RP < sup > 3 </ sup >, with fundamental group Z / 2, and only ( non-trivial ) covering space the hypersphere S < sup > 3 </ sup >, which is the group Spin ( 3 ), and represented by the unit quaternions.

Secondly the Lie algebra only determines uniquely the simply connected ( universal ) cover G * of the component containing the identity of a Lie group G. It may well happen that G * isn't actually a simple group, for example having a non-trivial center.

This is a topological constraint – there is no covering map from the 3-torus to the 3-dimensional real projective space ; the only ( non-trivial ) covering map is from the 3-sphere, as in the use of quaternions.

The quaternions H form a 4 dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals ( see below ).

This form generates the de Rham cohomology group meaning that any closed form is the sum of an exact form and a multiple of where accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane ( locally the derivative of a potential function ) being the derivative of a globally defined function.

A Lagrangian which involves only first derivatives in time formally leads to a zero Hamiltonian, but the Lagrangian itself may have non-trivial features which relate it to the topology of.

To break this further to the only non-trivial phenomenologically viable supersymmetry, N = 1, half of the supersymmetry generators must be projected out and this is achieved by applying the orientifold projection.

* and are the only non-trivial ( i. e. at least square ) consecutive powers of positive integers ( Catalan's conjecture ).

Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j ( κ ) ≥ λ.

A cardinal number κ is strongly λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ, j ( κ ) ≥ λ, and V ( λ ) is a subset of j ( M ).

Note that the two-dimensional case is the only non-trivial ( e. g. one dimension ) case where the rotation matrices group is commutative, so that it does not matter the order in which multiple rotations are performed.

In non-trivial cases, a full re-inspection is performed by the inspection team ( not only the moderator ).

Every quantum field theory satisfying certain technical assumptions about its S-matrix that has non-trivial interactions can only have a symmetry Lie algebra which is always a direct product of the Poincaré group and an internal group if there is a mass gap: no mixing between these two is possible.

Since the singleton coalitions ( consisting of a single lady ) are the only non-trivial coalitions of the game all possible distributions of this sum belong to the core, provided both ladies get at least € 5, the amount they can achieve on their own.

Here there may be an analogy with cancer: we can detect cancers by their rapidly accelerating growth, determinable only when related to the more normal rate of healthy growth.

It is good to know that Georgia will continue to have sufficient electrical power not only to meet the demands of normal growth but to encourage a more rapid rate of industrialization.

This stain often disrupts the normal cell activity or else colors only the outside.

In spite of normal thyroid function tests, a trial of propylthiouracil, 400 mg. daily for one week, was given but served only to intensify muscle weakness.

The necrosis often involved only a portion of the length of a given fiber, and usually the immediately adjacent fibers were normal.

The subjects were only given information about other possibilities of `` normal '' reaction.

It is simply that in Taoist tradition -- as in all good mysticisms -- books, words, or any other manifestations that belong to the normal state of consciousness are considered only the surface of experience.

Plus the fact that Siddo not only had the normal ( to Earthmen ) three genders of masculine, feminine, and neuter, but the two extra of inanimate and spiritual.

In their normal gait, only one leg is advanced at a time in the manner adopted by their ancestors, the lobe-finned fish.

Provided that an applet is hosted by an operating system, it can function as any other normal software application but is and performs only a small set of tasks.

For normal calculation a number zero is often needed, here most notably when calculating the number of years in a period that spans the epoch ; the end years need only be subtracted from each other.

trans-Cyclooctene is a stable strained alkene and the orbital misalignment is only 19 ° with a dihedral angle of 137 ° ( normal 120 °) and a degree of pyramidalization of 18 °.

The practice of only one bishop ordaining was normal in countries where the Church was persecuted under Communist rule.

It was normal Roman practice to allow allied kingdoms their independence only for the lifetime of their client king, who would agree to leave his kingdom to Rome in his will — the provinces of Bithynia and Galatia, for example, were incorporated into the Empire in just this way.

It was also suggested that individual genes are likely to have only a small effect and to be involved in some aspect related to the disorder ( and a broad range of " normal " human behavior ) rather than the disorder per se.

They are the only tiles in the whole set that don't match other tiles in the normal sense.

From 2007, every three-member states cooperate for their combined eighteen months on a common agenda, although only one formally holds the presidency for the normal six-month period.

A capital case could not be tried by a normal Beit Din of three judges, it can only be adjudicated by a Sanhedrin of a minimum of 23 judges.

It is not a vector field because it depends not only on the position of a particular material point, but also on the local orientation of the surface element as defined by its normal vector.

If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).

And since for normal files, two bytes of each physical sector were used by DOS as a pointer to the next physical track and sector of the file, only 254 out of the 256 bytes of a block were used for file contents.

Although a lens can precisely focus at only one distance at a time, the decrease in sharpness is gradual on each side of the focused distance, so that within the DOF, the unsharpness is imperceptible under normal viewing conditions.

This is huge when compared to normal renal sodium reabsorption which leaves only ~ 0. 4 % of filtered sodium in the urine.

In Corinth, the normal function of Ε to denote and was taken by a glyph resembling a pointed B ( x14px ), while Ε was used only for long close.

Maximum magnitudes along many normal faults are even more limited because many of them are located along spreading centers, as in Iceland, where the thickness of the brittle layer is only about 6 km.